A new conceptual approach for immersed boundaries based on volume-filtering

We present a novel approach to solving for the flow around moving immersed solids in a conservative manner without the need for body-fitting meshes, termed Volume-Filtered Immersed Boundary (VF-IB) method. This approach is derived by volume-filtering the conservation equations which transforms boundary conditions at solid-fluid interfaces into body-forces that apply to the right-hand side of the filtered mass and momentum equations. This approach answers long-standing questions related to IBMs: 1) what is the significance of the internal flow within the IB obtained when solving the conservation equations everywhere in the domain, 2) what is the correct choice of Lagrangian marker volume and 3) how does the sharpness of the IB surface affect the solution. The VF-IB method is physically and mathematically rigorous and does not depend on any ad-hoc numerical considerations, allowing us to explicitly express the IB forcing terms without assuming any discretization scheme. We implement and show how to couple this method with a finite-volume fluid solver. We show how to perform interpolation and extrapolation procedures using symmetric, compact, and unitary filter kernels to get accurate forces at the solid-fluid interface. To validate the approach, and help answer the above listed questions, we solve several canonical test cases, with both static and moving IBs. The VF-IBM yields excellent agreement with experiments and body-fitted mesh simulations. Further, we discuss how the choice of filter kernel impacts the solution quality.